Targeting Nominal GDP or Prices:

Expectation Dynamics and the Interest Rate

Lower Bound∗

Seppo Honkapohja, Bank of Finland

Kaushik Mitra, University of Saint Andrews

April 22, 2013; preliminary, please do not quote

Abstract

We examine global dynamics under infinite-horizon learning in

New Keynesian models where monetary policy practices either price-

level or nominal GDP targeting. These regimes are compared to in-

flation targeting. All of the interest-rate rules are subject to the zero

lower bound. The targeted steady state is locally but not globally sta-

ble under adaptive learning. Robustness properties of the three rules

in learning adjustment are compared using criteria for volatility of in-

flation and output, sensitivity to the speed of learning parameter and,

most importantly, domain of attraction of the targeted steady state.

Nominal income targeting with transparency is on the whole the best

policy regime, and inflation targeting is fairly close second. Price-level

targeting is the least robust regime according to these criteria.

JEL Classification: E63, E52, E58.

Keywords: Adaptive Learning, Monetary Policy, Inflation Target-

ing, Zero Interest Rate Lower Bound

∗Any views expressed are those of the authors and do not necessarily reflect the viewsof the Bank of Finland.

1

1 Introduction

The practical significance of the zero lower bound (ZLB) for policy interest

rates has become evident in the US and Europe since the 2007-9 financial

crisis and earlier in Japan since the mid 1990s. In monetary economics

the Japanese experience initiated renewed interest in ways of getting out

or avoiding the ZLB constraint.1 The long-standing Japanese experience

of persistently very low inflation and occasional deflation took place nearly

concurrently with the rise of inflation targeting as the most common, “best

practice” monetary policy since the early 1990’s. The ongoing economic

crisis in most advanced market economies and the already lengthy period

of very low interest rates has recently led to suggestions that price-level or

nominal GDP targeting can potentially be more appropriate frameworks for

the conduct of monetary policy.

History-dependence is a key feature of nominal income and price-level

targeting as it can provide more guidance to the economy than inflation

targeting.2 This guidance can be helpful and is arguably also good policy

in a liquidity trap where the ZLB is a constraint on monetary policy. See

Eggertsson and Woodford (2003) for a discussion of price-level targeting.

Carney (2012) and Evans (2012) provide broad discussions of the need for

additional guidance for the price level and possibly other variables including

nominal income or GDP. Carney (2012) suggests that with policy rates at

ZLB “there could be a more favorable case for nominal GDP targeting”.

Theoretical analysis of the ZLB as a constraint on interest rate policy

has recently been pursued in an inflation targeting framework that employs

a Taylor-type interest rate rule. It has been shown that there necessarily exit

multiple equilibria because the rule must respect the ZLB. Reifschneider and

Williams (2000) pointed out the existence of two steady states and discussed

other implications of ZLB using a backward-looking model with a piecewise

linear Taylor rule. Benhabib, Schmitt-Grohe, and Uribe (2001) showed that

in a New Keynesian (NK) model with a continuous Taylor rule there are

two steady states, the targeted steady state ∗ as well as a low-inflation (oreven deflationary) steady state . In addition, there can be a continuum

1For prominent early analyzes, see for example Krugman (1998), Eggertsson and Wood-

ford (2003), and Svensson (2003).2Price level targeting has received a fair amount of attention, see for example Svensson

(1999), Vestin (2006). Nominal income targeting has occasionally been considered, for

example see Hall and Mankiw (1994) and Jensen (2002) and the cited references.

2

of perfect foresight paths, starting from an initial ∗, which convergeasymptotically to . Using perfect foresight analysis, the multiple equilibria

issue was studied further in Benhabib, Schmitt-Grohe, and Uribe (2001) and

Benhabib, Schmitt-Grohe, and Uribe (2002).

It is important to observe that the possibility of multiple steady states has

broader applicability than inflation targeting and Taylor rules. In a standard

NK model the Fisher equation = −1, where is the gross interest rate,

is the subjective discount factor and is the gross inflation rate, is a key

equation for a nonstochastic steady state. Usually the interest rate rule has

a specified target inflation rate ∗ ≥ 1 (and an associated output level) as a

steady state. It is also possible that at a sufficiently low (or even less than

unity) inflation rate ≈ , together with the interest rate is very close to or

at one, and then the Fisher equation again holds and a second steady state

may thus exist. For example, if the rule specifies = 1 for in the interval

containing the value = , then , = 1 = −1 is a second steady

state. This argument does not require that a Taylor rule describes the policy

regime.3 As will be seen below, the existence of multiple steady states also

applies to standard versions of nominal GDP and price-level targeting.

The arguments in favor of nominal GDP or price-level targeting and the

cited studies about multiple equilibrium under a Taylor rule are both firmly

based on the rational expectations (RE) hypothesis. This standard approach

makes strong assumptions about the agents’ knowledge of the economy. The

agents with RE are able to predict the future path of the economy, except for

the consequences of unforecastable random shocks. The RE approach can be

questioned especially in global nonlinear settings that include an unfamiliar

region of outcomes that are caused by the ZLB. An alternative viewpoint

to RE, which assumes that agents’ economic knowledge have some form of

imperfection, has received rising support in recent years. Imperfect knowl-

edge has been specified in different ways in the recent literature. Probably

the most popular formulation has been the adaptive learning approach in

which agents maximize anticipated utility or profit subject to expectations

that derived from an econometric model. The model is updated over time as

new information becomes available.4

3Eggertsson and Woodford (2003), pp.193-194 note that such a deflationary trap can

exist but they do not analyze it in detail.4For discussion and analytical results concerning adaptive learning in a wide range of

macroeconomic models, see for example Sargent (1993), Evans and Honkapohja (2001),

Sargent (2008), and Evans and Honkapohja (2009).

3

The adaptive learning approach has been applied to the multiple equi-

libria issue caused by ZLB with a Taylor rule in some papers, see Evans

and Honkapohja (2005), Evans, Guse, and Honkapohja (2008), Evans and

Honkapohja (2010), and Benhabib, Evans, and Honkapohja (2012). In this

paper we use a standard nonlinear NKmodel to discuss nominal GDP target-

ing and price-level targeting from the viewpoint provided by global nonlinear

analysis. We argue that the good properties of nominal GDP or price-level

targeting advocated in the studies relying on the RE assumption no longer

hold under when imperfect knowledge prevails and agents behave as sug-

gested by adaptive learning.5 These targeting regimes can have good prop-

erties near the normal steady state ∗ but the regimes do not escape theproblems caused by the ZLB.

We also compare key properties of nominal GDP targeting to price-level

and also to inflation targeting. One of the main results is that nominal GDP

and inflation targeting are clearly more robust than price-level targeting to

deal with imperfect knowledge and learning. The second main result is that

commitment and transparency to the policy regime are helpful as they add

to robustness of the policy regimes in ways described below.

2 A New Keynesian Model

We employ a standard New Keynesian model as the analytical framework.6

There is a continuum of household-firms, which produce a differentiated con-

sumption good under monopolistic competition and price-adjustment costs.

There is also a government which uses monetary policy, buys a fixed amount

of output, and finances spending by taxes and issues of public debt as de-

scribed below.

The objective for agent is to maximize expected, discounted utility

5Williams (2010) makes a similar argument about price-level targeting under imperfect

knowledge and learning. His work relies on simulations of a linearized model.6The same economic framework is developed in Evans, Guse, and Honkapohja (2008).

It is also employed in Evans and Honkapohja (2010) and Benhabib, Evans, and Honkapohja

(2012).

4

subject to a standard flow budget constraint:

0

∞X=0

µ

−1

−1

− 1

¶(1)

+ + + Υ = −1−1 + −1

−1 −1 +

(2)

where is the consumption aggregator, and denote nominal and

real money balances, is the labor input into production, denotes the

real quantity of risk-free one-period nominal bonds held by the agent at the

end of period , Υ is the lump-sum tax collected by the government, −1

is the nominal interest rate factor between periods − 1 and , is the

price of consumption good , is output of good , is the aggregate

price level, and the inflation rate is = −1. The subjective discount

factor is denoted by . The utility function has the parametric form

=1−1

1− 1

+

1− 2

µ−1

¶1−2

− 1+

1 + −

2

µ

−1

− 1

¶2

where 1 2 0. The final term parameterizes the cost of adjusting

prices in the spirit of Rotemberg (1982). We use the Rotemberg formulation

rather than the Calvo model of price stickiness because it enables us to study

global dynamics in the nonlinear system.7 The household decision problem

is also subject to the usual “no Ponzi game” (NPG) condition.

Production function for good is given by

=

where 0 1. Output is differentiated and firms operate under monopo-

listic competition. Each firm faces a downward-sloping demand curve given

by

=

µ

¶−1

(3)

Here is the profit maximizing price set by firm consistent with its

production . The parameter is the elasticity of substitution between

two goods and is assumed to be greater than one. is aggregate output,

which is exogenous to the firm.

7The linearizations at the targeted steady state are identical for the two approaches.

5

The government’s flow budget constraint in real terms is

+ + Υ = + −1−1 + −1

−1 −1 (4)

where denotes government consumption of the aggregate good, is the

real quantity of government debt, and Υ is the real lump-sum tax collected.

We assume that fiscal policy follows a linear tax rule for lump-sum taxes as

in Leeper (1991)

Υ = 0 + −1 (5)

where we will assume that −1 − 1 1. Thus fiscal policy is “passive”

in the terminology of Leeper (1991) and implies that an increase in real

government debt leads to an increase in taxes sufficient to cover the increased

interest and at least some fraction of the increased principal.

We assume that is constant and given by

= (6)

From market clearing we have

+ = (7)

2.1 Optimal decisions for private sector

As in Evans, Guse, and Honkapohja (2008), the first-order conditions for an

optimum yield

0 = − +

( − 1)

1

(8)

+

µ1− 1

¶

1

(1−1)

−1 −

1

(+1 − 1)+1

−1 =

¡−1+1

−1

+1

¢and

= ()12

á1−−1

¢−1

2−1+1

!−12

where +1 = +1. We now make use of the representative agent

assumption. In the representative-agent economy all agents have the same

6

utility functions, initial money and debt holdings, and prices. We assume

also that they make the same forecasts +1 +1, +1, as well

as forecasts of other variables that will become relevant below. Under these

assumptions all agents make the same decisions at each point in time, so that

= , = , = and = , and all agents make the same

forecasts. Imposing the equilibrium condition = = one obtains the

equations

( − 1) =

µ −

µ1− 1

¶−1 −1

¶+

[(+1 − 1)+1]

−1 =

¡−1+1

−1

+1

¢

= ()12

á1−−1

¢−1

2−1+1

!−12

For convenience we assume that the utility of consumption and of money

is logarithmic (1 = 2 = 1). It is also assumed that agents have point expec-

tations, so that their decisions depend only on the mean of their subjective

forecasts. This allows us to write the system as

= (1−−1 )−1 (9)

−1 = +1(

+1)

−1, where +1 = +1, and (10)

( − 1) =

µ −

µ1− 1

¶−1 −1

¶+

£¡+1 − 1

¢+1

¤

(11)

Equation (11) is the nonlinear New Keynesian Phillips curve describing the

optimal price-setting by firms. The term ( − 1) arises from the quadratic

form of the adjustment costs, and this expression is increasing in over the

allowable range ≥ 12 To interpret this equation, note that the bracketed

expression in the first term on the right-hand side is the difference between

the marginal disutility of labor and the product of the marginal revenue from

an extra unit of labor with the marginal utility of consumption. The terms

involving current and future inflation arise from the price-adjustment costs.

Equation (10) is the standard Euler equation giving the intertemporal

first-order condition for the consumption path. Equation (9) is the money

demand function resulting from the presence of real balances in the utility

function. Note that for our parameterization, the demand for real balances

becomes infinite as → 1.

7

We proceed to rewrite the decision rules for and so that they depend

on forecasts of key variables over the infinite horizon (IH). The IH learning

approach in New Keynesian models was emphasized by Preston (2005) and

Preston (2006), and was used in Evans and Honkapohja (2010) and Benhabib,

Evans, and Honkapohja (2012) to study the properties of a liquidity trap.

2.2 The infinite-horizon Phillips curve

Starting with (11), let

= ( − 1) (12)

The appropriate root for given is ≥ 12and so we need to impose ≥

−14to have a meaningful model. Making use of the aggregate relationships

= 1 and = − we can rewrite (11) as

=

(1+) − − 1

( − )

−1 + +1

Solving this forward with = we obtain

=

(1+) − − 1

( − )−1 + (13)

∞X=1

−1¡+

¢(1+) − − 1

∞X=1

µ

+

+ −

¶≡ (

+1

+2)

where government spending is assumed to be constant over time. The ex-

pectations are formed at time and variables at time are assumed to be in

the information set of the agents. We will treat (13), together with (12), as

the temporary equilibrium equations that determine given expectations

{+}∞=1.

One might wonder why inflation does not also depend directly on the

expected future aggregate inflation rate in the Phillip’s curve relationship

(13).8 Equation (8) is obtained from the first-order conditions using (3) to

eliminate relative prices. Because of the representative agent assumption,

each firm’s output equals average output in every period. Since firms can be

assumed to have learned this to be the case, we obtain (13).9

8There is an indirect effect of expected inflation on current inflation via current output.9An alternative procedure would be to start from (8), iterate it forward and use the

8

2.3 The consumption function

To derive the consumption function from (10) we use the flow budget con-

straint and the NPG condition to obtain an intertemporal budget constraint.

First, we define the asset wealth

= +

as the sum of holdings of real bonds and real money balances and write the

flow budget constraint as

+ = −Υ + −1 + −1 (1−−1)−1 (14)

where = −1. Note that we assume () = , i.e. the rep-

resentative agent assumption is being invoked. Iterating (14) forward and

imposing

lim→∞

(+)

−1+ = 0 (15)

where

+ =

+1

Y=2

+−1

+

with + = +−1+, we obtain the life-time budget constraint of the

household

0 = −1 + Φ +

∞X=1

(+)

−1Φ+ (16)

= −1 + − +

∞X=1

(+)

−1(+ − +) (17)

where

Φ+ = + −Υ

+ − + + (+)−1(1−

+−1)+−1 (18)

+ = Φ+ + + = + −Υ

+ + (+)−1(1−

+−1)+−1

demand function to write the third term on the right-hand side of (8) in terms of the

relative price. This would lead to a modification of (13) in which future relative prices

also appear, but using the representative agent assumption and assuming that firms have

learned that all firms set the same price each period, the relative price term would drop

out.

9

Here all expectations are formed in period , which is indicated in the notation

for + but is omitted from the other expectational variables.

Invoking the relations

+ =

+ (19)

which is an implication of the consumption Euler equation (10), we obtain

(1−)−1 = −1 +−Υ+−1(1−−1)−1 +

∞X=1

(+)

−1+ (20)

As we have + = + −Υ+ + (+)

−1(1−+−1)

+−1, the final term

in (20) is

∞X=1

(+)

−1(+ −Υ+) +

∞X=1

(+)

−1(+)−1(1−

+−1)+−1

and using (9) we have

∞X=1

(+)

−1(+)−1(1−

+−1)+−1

=

∞X=1

(+)

−1(+)−1(−

+−1+−1) = −

1−

We obtain the consumption function

1 +

1− = −1 +

−1

+ −Υ +

∞X=1

(+)

−1(+ −Υ+)

The preceding derivation of the consumption function assumes households

that do not necessarily act in a Ricardian way, i.e. they do not impose the

intertemporal budget constraint (IBC) of the government. To simplify the

analysis, we assume that consumers are Ricardian, which allows us to modify

the consumption function as in Evans and Honkapohja (2010).10 From (4)

10Evans, Honkapohja, and Mitra (2012) state the assumptions under which Ricardian

Equivalence holds along a path of temporary equilibria with learning if agents have an

infinite decision horizon.

10

one has

+ + Υ = + −1−1 + −1 or

= ∆ + −1 where

∆ = −Υ − + −1−1

By forward substitution, and assuming

lim→∞

+ + = 0 (21)

we get

0 = −1 + ∆ +

∞X=1

−1+∆+ (22)

Note that ∆+ is the primary government deficit in + , measured as gov-

ernment purchases less lump-sum taxes and less seigniorage. Under the Ri-

cardian Equivalence assumption, agents at each time expect this constraint

to be satisfied, i.e.

0 = −1 + ∆ +

∞X=1

(+)

−1∆+ where

∆+ = −Υ

+ −+ +

+−1(+)

−1 for = 1 2 3

A Ricardian consumer assumes that (21) holds. His flow budget con-

straint (14) can be written as:

= −1 + , where

= −Υ − − + −1 −1

The relevant transversality condition is now (21). Iterating forward and using

(19) together with (21) yields the consumption function

= (1− )

à − +

∞X=1

(+)

−1(+ − )

! (23)

For further details see Evans and Honkapohja (2010).

11

3 Temporary Equilibrium and Learning

We assume that agents form expectations using steady-state learning, which

is formulated as follows. Steady-state learning with point expectations is

formalized as

+ = for all ≥ 1 and = −1 + (−1 − −1) (24)

for = . Here is called the “gain sequence,” and measures the

extent of adjustment of estimates to the most recent forecast error. In sto-

chastic systems one often sets = −1 and this “decreasing gain” learning

corresponds to least-squares updating. Also widely used is the case = ,

for 0 ≤ 1, called “constant gain” learning. In this case it is usually

assumed that is small.

It is worth emphasizing here that the notation for expectations refers to

future periods (and not the current one). This is important when computing

the stability conditions for learning (E-stability). The temporary equilibrium

equations with steady-state learning are:

1. The aggregate demand

= + (−1 − 1)( − )

µ

¶µ

−

¶(25)

≡ (

)

Here it is assumed that consumers make forecasts of future nominal

interest rates, which are equal for all future period, given that we are

assuming steady-state learning.

2. The nonlinear Phillips curve

= −1[( +1

+2)] ≡ −1[(

)] (26)

where () is defined in (13) and

() ≡ ( − 1) (27)

( ) ≡

µ−1

(1+) − ¡1− −1

¢

( − )

¶(28)

+

µ(1− )−1

µ−1( )

(1+) − ¡1− −1¢ −

¶¶

12

3. Bond dynamics

+ = −Υ +−1

−1 +

−1

(29)

4. Money demand

=

− 1 (30)

5. Different interest rate rules are considered below and are specified in

the next section.

The state variables are −1, −1, and −1. With Ricardian consumers

the dynamics for bonds and money do not influence the dynamics of the

endogenous variables, though clearly the evolution of and is influenced

by the dynamics of inflation and output. The system in general has three

expectational variables: output , inflation , and the interest rate

. The

evolution of expectations is given by

= −1 + (−1 − −1) (31)

= −1 + (−1 − −1) (32)

=

−1 + (−1 −−1) (33)

Analysis of E-stability is a convenient way to analyze conditions for con-

vergence of these kinds of learning models.11 Below we derive E-stability and

instability results for the steady states in this model. The E-stability con-

ditions of RE equilibrium give conditions for local convergence of real-time

learning rules such (31)-(33). The E-stability conditions apply directly for

rules that employ a decreasing gain but the same conditions also apply to

constant-gain rules in the limit, where the gain parameter is made arbitrarily

small.

4 Monetary Policy Frameworks

4.1 Price-level targeting

Starting with price-level targeting (PLT), we first note that a number of dif-

ferent formulations exist in the literature. We consider a simple formulation,

11See e.g. Evans and Honkapohja (2001) and Evans and Honkapohja (2009) for details

on E-stability analysis and its connections to real-time learning.

13

where the policy maker sets the policy instrument so that the actual price

level moves gradually toward a targeted price-level path, which is specified

exogenously. We assume that this target price-level path involves constant

inflation, so that

−1 = ∗ ≥ 1 (34)

In the “gradualist” formulation the interest rate, which is the policy instru-

ment, is set above (below, respectively) the targeted steady-state value of

the instrument when the actual price level is above (below, respectively) the

targeted price-level path , as measured in percentage deviations. The in-

terest rate also responds to the percentage gap between targeted and actual

levels of output. This idea leads to a Wicksellian interest rate rule

= 1 + max[− 1 + [( − )] + [( − ∗)∗] 0] (35)

where the max operation takes account of the ZLB on the interest rate and

= −1∗ is the gross interest rate at the targeted steady state.12 For

analytical ease, we adopt a piecewise linear formulation of the interest rate

rule.

4.2 Nominal GDP targeting

A standard formulation of nominal GDP targeting obtains from postulating

that the monetary authority sets the interest rate in each period to meet a

targeted nominal GDP path in each period. In line with a standard NK

model, assume also that there is an associated inflation objective calling for

a non-negative (net) inflation rate and that the economy does not have any

sources for trend real growth. Then formally, = and −1 = ∗, andwe obtain the commonly used concept of nominal GDP growth targeting, see

e.g. Mitra (2003) and the references therein. Taking time differences,

− −1 = ∆ = ∗ (36)

Requiring that actual growth of nominal GDP equals the target, i.e.,

−1−1

= ∆

12See pp.260-61 of Woodford (2003) and Giannoni (2012) for discussion of Wicksellian

rules.

14

yields the equation

∆−1 =

which can be coupled to the temporary equilibrium equations (25) and (27)-

(28). We obtain the equation

∆−1

= (

) (37)

Clearly, the right-hand side of (37) is decreasing in , so that a unique

interest rate in can be computed. When the right-hand side takes the specific

form (25), it is possible to compute an explicit form for the interest rate .

It is given by

= (−1 − 1)(

− )

∆−1 −

µ

−

¶≡ (

−1) (38)

so that the current interest rate depends on expected inflation, expected

future interest rate and also on past gross and expected net output.

The formulation (38) is, however, incomplete as it does not take into ac-

count the ZLB. The preceding solution is meaningful only when it does not

violate the ZLB. If the ZLB constraint is binding in temporary equilibrium,

then the monetary authority cannot meet the nominal GDP target. The con-

strained temporary equilibrium is then determined by = 1 and equations

(25) and (27)-(28). Incorporating the constraint, the interest rate rule takes

the final form

= max[1 (

−1)] (39)

5 Steady States

A non-stochastic steady state ( ) under PLT must satisfy the Fisher

equation = −1, the interest rate rule (35), and steady-state form of the

equations for output and inflation (25) and (26). One steady state clearly

obtains when the actual inflation rate equals the inflation rate of the price-

level target path, see equation (34). Then = ∗, = ∗ and = ∗, where∗ is the unique solution to the equation

∗ = −1[( (∗ ∗ ∗ ∗) ∗)]

Moreover, for this steady state = for all .

15

Then consider the possible steady states under nominal GDP targeting.

One of the steady states obtains when the economy follows the targeted

nominal GDP path, so that = ∗, = ∗ and = ∗ as under PLT and∗ = ∆.

The targeted steady state under either PLT or nominal GDP targeting

is, however, not unique.13 It can be verified that there is a second steady

state in which the ZLB condition is binding:

Lemma 1 Assume that −1∗ − 1 . There exists a ZLB-constrained

steady state under PLT in which = 1, = , and solves the equation

= −1[( ( 1 1) )] (40)

Correspondingly, a ZLB-constrained steady state exists nominal GDP target-

ing. In this steady state the price-level target or nominal GDP target ∆,

respectively, is not met.

Proof. Consider the interest rate rule (35). Imposing = 1 implies

that → 0 while → ∞ (or if ∗ = 1) as → ∞. It follows that − 1 + [( − )] + [( − ∗)∗] 0 for sufficiently large when

→ ∗, so that = 1 in the interest rate rule. A unique steady state

satisfying (40) is obtained. Thus, , and constitute a ZLB-constrained

steady state.

Now consider the economy under nominal GDP targeting and impose

= 1, = , and = where solves (40) with = . As just noted, these

requirements yield a steady state state for the economy. Moreover, setting

expectations and actual values of ( ) equal to ( 1) the equation (38)

does not hold and

( 1 ) 1

so that the growth of nominal GDP is below the target. Yet, ( ) =

( 1) is a ZLB-constrained steady state as the complete interest rate rule

(39) holds.

We remark that the sufficient condition −1∗− 1 is not restrictive

e.g. when = 099 and ∗ = 102.14

13Existence of the two steady states under PLT was pointed out in Evans and Honkapo-

hja (2013), section 2.5.3. This paper also considers E-stability of the steady states under

short-horizon (called Euler equation) learning.14A weaker sufficient condition is −1∗− 1−+(

∗− 1) 0, in which the term∗is complicated function of all model parameters.

16

The lemma states that the commonly used formulations of price-level and

nominal GDP targeting both suffer from global indeterminacy as the economy

has two steady states under either monetary policy regime. We next consider

global dynamics of the economy in these regimes under the hypothesis that

agents operate under imperfect knowledge and their expectations are formed

in accordance with steady-state learning as described above.

6 Expectations Dynamics: Theoretical Re-

sults

6.1 Price-level targeting

We begin to analyze the dynamics of learning under price-level targeting

formulated in Section (4.1) above. Considering the Wicksellian rule, we have

the system (25), (26), (35) and (34), together with the adjustment of ex-

pectations given by (31), (32) and (33). Since the model is nonstochastic,

we impose steady-state learning as specified by the adjustment equations for

expectations.15

The first step is to consider local dynamics near the two possible steady

states. Here some theoretical results can be obtained by exploiting the E-

stability method that is commonly used in the study of adaptive least-squares

learning. However, for most part it is necessary to employ simulations of this

system to obtain aspects of global properties of the economy under price-

level targeting and with expectations formed by steady-state learning. In

the numerical analyses we treat a general case and allow for trend inflation,

i.e. ∗ 1.

In the special case ∗ = 1 there is no explosive exogenous time-dependence

and it is possible to partial analytical results of the local stability properties

of the two possible steady states under adaptive learning. For this we write

the temporary equilibrium system (25), (26), (35) and (34) in the abstract

form

( −1) = 0

15Preston (2008) discusses local learnability of the targeted steady state with IH learning

when the central bank employs PLT. In the earlier literature Evans and Honkapohja (2006)

and Evans and Honkapohja (2013) consider E-stability of the targeted steady state under

Eular equation learning for versions of PLT.

17

where the vector = ( ) . Linearizing around a steady state we

obtain

= (−)−1(

+ −1

−1) ≡ + −1 (41)

where for brevity we use the same notation for the deviations for the steady

state. Recall that refers to the expected future values of and not the

current one. Below it is assumed that information about the current value

of endogenous variables is not available when expectations in period are

formed. This system is in a standard form for the analysis of E-stability of

RE equilibrium.

Consider the possible RE solutions of the form

= −1 (42)

for the system (41).16 It can be shown that solves the matrix equation

= ( −)−1 (43)

The E-stability condition is that all eigenvalues of the matrices

( + ) and 0 ⊗ + ⊗

have real parts less than one.

In general, the analytical details for E-stability of PLT appear to be

intractable, but results are available in the limiting case → 0, i.e. price

adjustment costs are sufficiently small. Moreover, we make the normalization

= 1 for simplicity. It is possible to obtain the following result:17

Proposition 2 Assume ∗ = 1 and → 0. Then under PLT the targeted

steady state with = 1 and = −1 is E-stable if 0.

16We note here that the formulation of E-stability outlined below allows for learning

of both intercepts and AR-coefficients of agents’ perceived law of motion. Complete E-

stability conditions are then obtained. In contrast, analysis of real-time learning specified

in (31)-(33) is restricted to learning of the steady state values because, for simplicity, the

model does not include random shocks. See Evans and Honkapohja (1998) for method-

ological issues about modeling learning in deterministic and stochastic frameworks.17Mathematica routines containing technical derivations in the proof are available upon

request from the authors.

18

Proof. First consider the possible RE solutions of the form (42). Since only

the lagged price level influences the RE dynamics, the first three columns of

matrix are zero vectors. Moreover, in the limit → 0 it can be shown that

the equation for becomes

=

− 1

so that the movement of under learning influences other variables but not

vice versa. The dynamics of and +1 converge to the steady state as 1.

From this it follows that the (4 1) element of is equal to zero, i.e. in what

follows we can impose that 14 = 0.

In the limit → 0 we have

=

⎛⎜⎜⎜⎜⎝

−10 0 0

2(−∗)−∗(−∗)(−1)

−1(−1)

(−1)0

1(−∗)(−1)

−1(−1)

−10

∗+2(∗−)

(−∗)(−1)

−1(−1)

(−1)0

⎞⎟⎟⎟⎟⎠ , =

⎛⎜⎜⎝0 0 0 0

0 0 0 −1

0 0 0 0

0 0 0 0

⎞⎟⎟⎠

The MSV solutions for the system are given by equation (43), where

=

⎛⎜⎜⎝0 0 0 0

0 0 0 240 0 0 340 0 0 44

⎞⎟⎟⎠

It can be shown that the unknown coefficients 24, 34 and 44 can be solved

from the system

24 = −1 34 = 0 and 44 = 0

Next, calculate the eigenvalues of matrices ( + ) − and 0 ⊗ +

⊗− , where is a steady state. The eigenvalues of ( + ) are

−1−11

− 1

1

− 1

and the eigenvalues of 0 ⊗ + ⊗ consist of 12 roots equal to −1 and

four roots equal to

−1− 1

(1− )

19

so that the MSV solution is E-stable whenever 0

We emphasize that by continuity of eigenvalues, E-stability of the targeted

steady state also obtains when is sufficiently small. Below we carry out

numerical simulations for other parameter configurations and with strictly

increasing price-level target. The learning dynamics do converge for many

cases with non-zero value of .

As regards the other steady state in which ZLB binds, we have the result:

Proposition 3 Assume ∗ = 1. The steady state with binding ZLB ( 1 )

is not E-stable under PLT.

Proof. When the ZLB binds, the interest rate is constant and the price

level evolves exogenously in a neighborhood of the constrained steady state.

The temporary equilibrium system and learning dynamics then reduce to

two variables and together with their expectations. Moreover, no lags

of these variables are present, so that the abstract system (41) is two dimen-

sional with = ( ) and = 0.

It can be shown that

( − ) =(1+)(1 + )( − )2 + 2( − 1)

( − )2( − 1)2(2 − 1)

The numerator is positive whereas the denominator is negative. Thus,(−) 0, which implies E-instability.

The preceding results show that the targeted steady state is locally but

not globally stable under learning. In Section 7 we present a variety of numer-

ical results for PLT. These results do not employ the restrictive assumption

of a small that was invoked in Propositions 2 and 3.

6.2 Nominal GDP targeting

In this section, we consider the consequences of nominal GDP targeting under

learning. We have the system (25), (26), (36) and (38), together with the

adjustment of expectations given by (31), (32) and (33).

We first consider local dynamics near the two possible steady states by

using the E-stability technique on the linearization (41). The state variable

is now = ( ) . The following result holds for the targeted steady

state:18

18Mitra (2003) discusses E-stability of the targeted steady state under Euler equation

learning.

20

Proposition 4 Assume → 0. Then there exists 0 0 such that for

∈ [0 0) targeted steady state with ∗ = 1 and = −1 is E-stable under

nominal GDP targeting.

Proof. We consider the RE solution of the form (42) and the E-stability

conditions that all eigenvalues of the matrices( + ) and 0⊗ + ⊗

have real parts less than one. In the case → 0 the coefficient matrices are

given by

=

⎛⎜⎝

−10 0

−(−∗)2((−1+∗+)−∗(−2+2∗+))

(−∗)2∗2(−1)

(1−∗)+∗2(−2+∗+)

(−∗)∗(−1)

(∗−1)(−∗2)2

(−∗)∗(−1)−+2∗

(−∗)∗(−1)−1

(−1)1

−1

⎞⎟⎠ , =

⎛⎝ 0 0 01∗ 0 0

0 0 0

⎞⎠

From these matrices it is evident that the matrix has second and third

columns equal to zero in the RE equilibrium, i.e. it can be assumed that

=

⎛⎝ 11 0 0

12 0 0

13 0 0

⎞⎠when solving for using (43). Given and , we get 11 = 0, 12 = 1

∗ , and

13 = 0.

To assess E-stability, the eigenvalues of ( + ) are

−1 + (∗ − 2)∗2

( − ∗)∗( − 1)

1

− 1

while all eigenvalues of 0⊗ + ⊗− are equal to −1. All eigenvalues

except+(∗−2)∗2

(−∗)∗(−1)are clearly negative. We need to examine this remaining

eigenvalue in more detail.

From (27)-(28) one can obtain the steady-state relation between ∗ and

−1(∗)(1+) − ¡1− −1¢ ∗

(∗ − )= 0 (44)

21

We note that ∗ 1 when = 0 and then the remaining eigenvalue is

negative. From equation (44) one can solve in terms of ∗. We obtain

= −1(1− )(∗)1−(1+) + ∗

which is an increasing function in (∗ )-space and at point ∗ = 1 it has a

value 1 + −1( − 1) 1. Requiring that the value of the numerator of the

eigenvalue be negative yields the restriction

(2− ∗)∗2

The right-hand side of the last formula is increasing for ∗ 43 and de-

creasing for ∗ 43. At ∗ = 1 the right-hand side equals 1. These

Considerations imply the existence of the asserted 0 0.

Recall that the ZLB-constrained steady state also exists under nominal

GDP targeting. One then has = 1 in the implied interest rate rule (39).

The analysis of E-stability of this steady state is formally the same as in

Proposition 3, and one has:

Proposition 5 Assume ∗ = 1. The steady state with binding ZLB ( 1)

is not E-stable under nominal GDP targeting.

7 Numerical Analysis: Non-transparent Pol-

icy

7.1 Dynamics under PLT

We adopt the following calibration: = 25, ∗ = 102, = 099, = 07,

= 350, = 21, = 1, and = 02. We choose = 25 to clearly separate

the intended and unintended steady states in the numerical analysis. Our

results are robust to using = 15, which is the usual value for the interest

rate rule. The calibrations of ∗ and are standard. We set the

labor supply elasticity = 1 The value of = 21 was chosen so that the

implied markup of prices over marginal cost at the steady state is 5 percent,

which is consistent with the evidence presented by Basu and Fernald (1997).

Following Sbordone (2002), we set = −175(1 + ) = 350. It is also

assumed that interest rate expectations + = +−1+ revert to the

steady state value −1 for ≥ . We use = 28 which under a quarterly

22

calibration corresponds to 7 years. To facilitate the numerical analysis the

lower bound on the interest rate is set slightly above 1 at value 1001. The

gain parameter is set at = 0002, which is a low value. Sensitivity of this

choice is discussed below.

The targeted steady state is ∗ = 0944025, ∗ = 102 and the low steady

state is = 0942765, = 099. The initial conditions in the next sim-

ulation are set as follows: (0) = (0) = 0945, (0) = (0) = 1025,

(0) = (0) = ∗ and ((0) − (0))(0) = 103 The initial value for

((0) − ∗)∗ is determined by the value (0). For policy parameters weadopt the values = 200 and = 018 which approximate the values by

Giannoni (2012), Table 1 (with iid shocks).

Figures 1-3 show the dynamics inflation and the interest rate for PLT

from these initial conditions. It is seen that there is oscillatory convergence

to the targeted steady state. There are significant initial oscillations and

hits the lower bound almost 20% of the time periods despite assuming initial

conditions for inflation and output above the targeted steady state.19

100 200 300 400 500 t

1.00

1.05

1.10

1.15

p

Figure 1: Inflation dynamics under PLT

19For numerical purposes, we define the zero lower bound to be the interval when is

between 1 and 1005

23

100 200 300 400 500 t

0.7

0.8

0.9

1.0

y

Figure 2: Output Dynamics under PLT

100 200 300 400 500 t

1.2

1.4

1.6

1.8

R

Figure 3: Interest Rate Dynamics under PLT

We then carried out a few sensitivity analyzes. First, the effect of the

policy parameters and do not have a very big impact on the dynamics,

except that very low values for can lead to instability. Second, convergence

to the targeted steady state is surprisingly sensitive to the magnitude of gain

parameter. Raising the gain from the base value of = 0002 to = 0004

led to instability. If the target is unstable, then the dynamics will eventually

enter the deflationary region for which , and get trapped there.

This last observation raises questions of the robustness of PLT. It is worth

examining the robustness of PLT in terms of sensitivity to (i) initial condi-

tions, (ii) size of the gain parameter and (iii) magnitude of the fluctuations.

24

As a point of comparison we take the policy framework of inflation targeting

(IT) by means of a Taylor rule

= 1 + max[− 1 + [( − ∗)∗] + [( − ∗)∗] 0]

with the usual values = 15 and = 05, same initial conditions and the

same calibrated model.

Looking first at convergence from initial conditions, the top part of Table

1 looks at implications of variations in initial output and output expectations

(these two are kept equal to each other) when inflation and the interest

rate and their expectations are kept at the steady state. "conv" and "div"

indicate that there is convergence to (divergence from, respectively) to the

targeted steady state for PLT and IT from the indicated initial conditions.

The bottom half considers implications of variation in initial inflation and

inflation expectations (the two are kept equal to each other) when initial

output and output expectations are set at the steady-state value.

(0) = ∗ (0) = 094 0941 0942 0946 0948

div div conv conv div

div conv conv conv conv

(0) = ∗ (0) = 1008 101 103 104 105

div conv conv div div

div conv conv conv conv

Table 1: Partial domains of attraction of learning under PLT and IT

Table 1 shows that IT is more robust than PLT in this sense. The set of initial

conditions from which there is convergence to the targeted steady state seems

to be slightly larger for IT than for PLT.

Second, we consider robustness of IT and PLT with respect to the mag-

nitude of the gain parameter . The initial conditions were set at values

that were used to generate Figures 1-3. The range of for which there is

convergence to the targeted steady state under PLT is approximately equal

to (0 0003). The corresponding range for IT is approximately (0 0125).

Clearly, IT is much more robust than PLT in this sense.

The third comparison of robustness was done in terms of the range of

variation in inflation and output during the learning adjustment for the basic

25

simulation of Figures 1-3. The iterations were done for 2000 periods (with

the gain = 0002). For PLT the ranges are ∈ [0951 1175] and ∈[0561 1030]. For IT we obtained ∈ [1018 1041] and ∈ [0940 0945].

These results suggest that IT is more robust than PLT as it induces smaller

fluctuations in inflation and output than PLT.

7.2 Dynamics under nominal GDP targeting

We now consider aspects of learning dynamics under nominal GDP targeting.

For the basic simulation, we use the same calibration and initial conditions

as in the preceding section. The gain parameter is also set at the same value

= 0002 and below consequences of higher values for are considered.

Figures 4-6 show the evolution of inflation, output and the interest rate

under nominal GDP targeting.

100 200 300 400 500 t

1.020

1.025

1.030

1.035

p

Figure 4: Inflation dynamics under NGDP targeting

26

100 200 300 400 500 t

0.89

0.90

0.91

0.92

0.93

0.94

y

Figure 5: Output dynamics under NGDP targeting

100 200 300 400 500 t1.04

1.06

1.08

1.10

1.12

1.14

R

Figure 6: Interest rake under NGDP targeting

Comparing these figures to Figures 1-3, it is seen that under nominal

GDP targeting the movements of inflation, output and interest rate in the

early periods are oscillatory, which resembles dynamics under PLT. A major

difference between the two policy regimes is that the oscillations are more

persistent under PLT than under nominal GDP targeting. For the latter the

fluctuations die out fairly quickly. Both PLT and nominal GDP targeting

27

differ from IT in that the latter does not show any oscillations from the same

initial conditions (figures are not shown but are available).

Next, we examine the robustness of nominal GDP targeting with respect

to (i) the size of domain of attraction of the targeted steady state, (ii) gain

parameter and (iii) magnitude of the fluctuations. The comparison is made

to the IT regime, since IT performed better in the preceding contest between

IT and PLT.

Looking first at convergence from initial conditions, the top part of Table

2 looks at implications of varying in initial output and output expectations

(these two are kept equal to each other) when inflation and the interest rate

and their expectations are kept at the steady state. As before, "conv" and

"no conv" indicates that there is convergence (non-convergence, respectively)

to the targeted steady state from the indicated initial conditions. The bottom

half of the table considers implications of variations in initial inflation and

inflation expectations (the two are kept equal to each other) when initial

output and output expectations are set at the steady-state value.

(0) = ∗ (0) = 094 0941 092 0946 0948 0950

div conv conv conv conv div

div conv conv conv conv conv

(0) = ∗ (0) = 101 1011 103 104 105 106

conv conv conv conv conv conv

div conv conv conv conv conv

Table 2: Partial domains of attraction of learning under nominal GDP

targeting and IT

Table 2 shows that nominal GDP targeting and IT are nearly equally robust

in this sense. There are some differences, but they occur at some distance

away from the targeted steady state.

Second, we consider robustness of IT and nominal GDP targeting with

respect to the magnitude of the gain parameter . The initial conditions

were set at values that were used to generate Figures 4-6. The range of

for which there is convergence to the targeted steady state under nominal

GDP targeting is approximately equal to (0 018). The corresponding range

for IT is approximately (0 0105). The conclusion here is that nominal GDP

targeting is somewhat more robust than IT, though it should noted that the

28

ranges for giving convergence in both regimes are large and well cover the

estimated or calibrated values for the gain parameter that are used in the

applied literature.

The third comparison of robustness was done in terms of the range of

variation in inflation and output during the learning adjustment for the basic

simulation of Figures 4-6. The iterations were done for 2000 periods (given

that the gain was kept small). For nominal GDP targeting the ranges were

∈ [1015 1038] and ∈ [0881 0948]. Comparing these ranges to those

obtained above for IT, which are ∈ [1018 1041] and ∈ [0940 0945], it

is seen that for inflation fluctuations the ranges are close to each other. In

terms of output fluctuation, IT does clearly better. We can conclude that

IT is more robust than nominal GDP targeting in that it appears to induce

smaller fluctuations in output.

8 Transparency of Policy

A key underlying assumption in the preceding analysis is that the behavior

of the policy maker is not transparent, so that private agents must make

forecasts of future interest rates. This is a restrictive assumption and major

part of the recent discussions of the desirable properties of nominal GDP and

price-level targeting has relied on credibility and transparency of the policy

regime. However, it must be stressed that these earlier analyses additionally

rely on the RE hypothesis. As was already pointed out in the introduction,

it is important to consider the alternative assumption that expectations of

private agents are not rational.

Even if private agents’ knowledge is imperfect, credibility and trans-

parency of policy regime can be built into the model by postulating that

agents know the policy and use this knowledge in their forecasting of future

interest rates. In this kind of setting agents do not directly forecast the in-

terest rate but instead use the policy rule, so that agents use forecasts for

the future values of the variables in the policy rule.

In this section we first develop the details of the formulation of trans-

parent nominal GDP targeting. Then we compare its robustness properties

to the non-transparent formulation. Later, we compare robustness of trans-

parent nominal GDP targeting to the transparent forms of price-level and

IT targeting. For brevity, we omit formal details of the latter, as they are

straightforward: one simply substitutes the interest rate rule into the aggre-

29

gate demand equation (25).20 It should also be pointed out that for reasons

of brevity we also omit the theoretical analysis of E-stability of the steady

states under the different transparent policy regimes. Since the simulations

converge, E-stability should not be a concern.

Under nominal GDP targeting the policy rules is given (39). There are

two cases to consider for temporary equilibrium and learning. If ZLB is not

expected to bind, interest rate expectations are computed from

= (−1 − 1)

( − )

∆−1 −

µ

−

¶as current data is not used in expectations formation. Substituting this

expression and the rule for the actual interest rate (39) in the aggregate

demand function yields

= ∆−1

Applying (31) and (32), the learning dynamics are then

+1 = + ( − ) (45)

+1 = + (−1[( )]− (46)

where = ∆−1 for period + 1. If instead ZLB is expected to bind,

then = 1 and

= ( 1), where

= max

∙1 (−1 − 1)

( − )

∆−1 −

µ1

1−

¶¸

These equations are then coupled with temporary equilibrium for inflation

(27)-(28) and the learning system (45)-(46).

It turns out that transparency of the policy rule has a major impact

on the domain of attraction of the targeted steady state, so we consider it

first. This general result holds for the three interest rate rules considered,

but the improvement is biggest from nominal GDP and inflation targeting.

Transparency about the policy rule yields some improvement also for PLT

and lack of robustness of PLT relative to nominal GDP and inflation targeting

is accentuated. Table 3 presents the results.21

20For example, see Eusepi and Preston (2010) and Preston (2006).21The initial conditions and other parameter values were the same as those for Figures

1-3. In PLT simulation, the initial price and output deviations from their targets were set

at 3 percent.

30

(0) = ∗ (0) = 06 0935 0938 094 0942 094 0946 0948

conv conv conv conv conv conv conv conv

div div conv conv conv conv conv conv

div div conv conv conv conv conv conv

(0) = ∗ (0) = 0985 099 0991 1002 1003 102 104 105

div div conv conv conv conv conv conv

div conv conv conv conv conv conv conv

div div div div conv conv conv conv

Table 3: Partial domains of attraction for nominal GDP, inflation and

price-level targeting with transparent policy

The results show that in terms of domain of attraction nominal GDP tar-

geting is the best on the whole. IT is nearly as robust as nominal GDP

targeting, whereas PLT is clearly less robust. The most remarkable feature

for both nominal GDP and inflation targeting is that the lowest value for

the initial condition on (0) (and (0) = (0)) can be pushed to arbitrarily

close to the low steady state value = 099 in the partial domains shown.

Comparing to the results in Tables 1 and 2, it is seen that there is major

improvement to the corresponding regime without transparency of policy.

Considering at the partial domain of attraction in terms of (0), the im-

provement is huge for nominal GDP targeting but less dramatic for both IT

and PLT. The general conclusion here is that nominal GDP targeting is the

most robust policy regime. IT is somewhat less robust and PLT is clearly

least robust of the three rules considered according to this criterion.

Next, we look at the impact of policy transparency for the three rules

with respect to the other measures of robustness, namely speed of learning

(the gain parameter) and the range of variation. The results make use of the

same initial conditions as in the corresponding analysis in Section 7.22

The results for the maximal value of the gain parameter for the three

policy rules under transparency are:

0 0190

0 0126

0 0024

22Note that there is no need for an assumption about (0) in this section.

31

Table 4: Robustness with respect to the gain parameter with transparent

policy

It is seen that policy transparency improves robustness with respect to speed

of learning parameter for nominal GDP, PLT and inflation targeting. Com-

paring to the corresponding results with non-transparent policy, the changes

are rather small, except for PLT.

The results about volatility in terms of ranges for inflation and output

are as follows for robustness of the three policy rules under transparency:

∈ [1014 1038] ∈ [0892 0953]

∈ [1019 1039] ∈ [0929 0945]

∈ [0967 1148] ∈ [0667 1090]

Table 5: Robustness with respect to volatility with transparent policy

It is seen that IT performs the best in terms of inflation and output volatility.

Nominal GDP targeting has more variability for both variables then IT, while

performance of PLT is clearly the worst in terms of both inflation and output

volatility.

9 Conclusion

Our objective has been to provide an assessment of price-level and nomi-

nal GDP targeting that have been recently suggested as improvements over

inflation targeting policy. There are two important starting points for our

analysis.

First, it is assumed that agents have imperfect knowledge and therefore

their expectations need not be rational. Instead, agents make their forecasts

using an econometric model that is updated over time. It should be empha-

sized that our analysis covers both non-transparent and transparent policy

regimes within this approach.

Second, we have carefully introduced the nonlinear global aspects of a

standard framework, so that the implications of the interest rate lower bound

can be studied. It is well-known that inflation targeting with a Taylor rule

32

suffers from global indeterminacy and it was shown here that the same prob-

lem exists for both standard versions of price-level and nominal GDP tar-

geting. Theoretical results for local (but not global) stability under learning

of the targeted steady state were derived for the latter two policy rules23,

whereas the low inflation steady state is locally unstable under learning with

each policy rule.

We then examined further properties of the three policy rules under cases

of non-transparency or transparency of policy. The goal was to consider

different dimensions of robustness of the policies. The three criteria for ro-

bustness were as follows. First, sensitivity with respect to the speed of private

agents’ learning was studied, i.e. for what values of learning speed is there

convergence of learning dynamics to the targeted steady state. Second, we

looked at the range of fluctuation in inflation and output in a typical path of

learning adjustment for each of the three policy rules under non-transparency

and transparency. Third and most importantly, what is the domain of at-

traction of the desired steady state under the different rules, i.e., how big is

the set of initial conditions from which there is convergence to the desired

steady state.

The current numerical results are preliminary. A more systematic study

of especially the third criterion of robustness is complex in a multidimensional

dynamic system that as multiple steady states. Also the other criteria can

be done with more systematic methods and we will do this. The preliminary

results are, however, suggestive and surprising.

The results are surprising in our opinion. Overall, nominal GDP targeting

with transparency of the policy rule appears to have the best robustness

properties. Inflation targeting with transparency reaches the second place in

the comparison and its robustness is not that far below that of nominal GDP

targeting. For volatility of inflation and output, inflation targeting has the

smallest range of variation. Transparency about the policy rules is helpful

under both nominal GDP and inflation targeting. Price-level targeting is

clearly the least robust policy rule of the alternatives considered.

23The corresponding result for IT is known. See e.g. Evans and Honkapohja (2010) for

the forward-looking version of the Taylor rule.

33

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